Existence and nonexistence results for second-order Neumann boundary value problem

In this paper some existence and nonexistence results for positive solutions are obtained for second-order boundary value problem -u"+Mu=f(t,u), t∈(0,1) with Neumann boundary conditions u'(0)=u'(1)=0, where M>0, f∈C([0,1]×R+, R+). By making use of fixed point index theory in cones, some new results are obtained.